The purpose of this essay is to investigate the evolution of two ticking metronomes that lie on a light, movable surface, such as a skateboard. Any such system where the metronomes tick unsynchronized will eventually synchronize, so long as the frequencies of the metronomes is identical. This is due to a transferal of momentum from the pendulums through the movable surface. This results in a form of interference between the transferred momentums of the metronomes. This behavior is quite interesting to watch, and has spawned several viral videos.

My research question for this project is To what extent does the Kuramoto model provide a theoretical framework for a coupled system of metronomes? The Kuramoto model, named after Japanese physicist Yoshiki Kuramoto, provides a general differential equation for N coupled oscillators. Our skateboard scenario provides such a coupling for the metronomes. I adjust and solve the corresponding equation for N=2, obtaining a theoretical model for the evolution and eventual synchronization of the coupled system.

To test the model, I collected data of two metronomes synchronizing by recording the time lag between their ‘ticks’, which turned out to be a rather arduous affair. I did this with the audio software Audacity. Because the model does not predict a clean linear relationship, I transformed it through a function that my model predicted would yield a linear graph. The corresponding correlation was strong, and the data fit well. However, the model broke down as the phase difference approached very small values. The Kuramoto model was able to accommodate for this when I changed my assumption that the metronomes had identical frequencies (they did in fact not). With this correction, I conclude that the Kuramoto model provides a highly useful framework for the coupled metronome system consisting of two metronomes. However, more research is required to give a complete picture of the model’s application, including its use in multi-metronome systems.